Using a stand or hand-held mixer, beat together the butter and sugar on medium speed for 3 minutes. Add the egg and vanilla extract. Beat until just incorporated. In a separate bowl, combine the flour, cocoa powder, baking soda and salt. Gradually add to the butter mixture, mixing until just combined.
Shape dough into two disks and wrap in plastic wrap. Refrigerate for 2 hours.
Preheat oven to 350 degrees F. Line 2 baking sheets with parchment paper. On a lightly floured surface, roll the dough out to 1/4-inch thickness. Using 2-inch round cookie cutters, cut the dough into circles. Place on prepared baking sheets. Roll out excess dough and repeat process.
Bake in preheated oven for 6 to 8 minutes, or until cooked through in centers. Let cool. Sandwich dulce de leche between 2 cookies (use between 1/2 and 1 teaspoon for each cookie sandwich, depending on how runny the sauce is). Repeat process and refrigerate cookies until ready to serve!